eifueo/docs/1b/math119.md
2023-01-15 17:32:31 -05:00

4.1 KiB
Raw Blame History

MATH 119: Calculus 2

Multivariable functions

!!! definition - A multivariable function accepts more than one independent variable, e.g., \(f(x, y)\).

The signature of multivariable functions is indicated in the form [identifier]: [input type][return type]. Where \(n\) is the number of inputs:

\[f: \mathbb R^n \to \mathbb R\]

!!! example The following function is in the form \(f: \mathbb R^2\to\mathbb R\) and maps two variables into one called \(z\) via function \(f\).

$$(x,y)\longmapsto z=f(x,y)$$

Sketching multivariable functions

!!! definition - In a scalar field, each point in space is assigned a number. For example, topography or altitude maps are scalar fields. - A level curve is a slice of a three-dimensional graph by setting to a general variable \(f(x, y)=k\). It is effectively a series of contour plots set in a three-dimensional plane. - A contour plot is a graph obtained by substituting a constant for \(k\) in a level curve.

Please see level set and contour line for example images.

In order to create a sketch for a multivariable function, this site does not have enough pictures so you should watch a YouTube video.

!!! example For the function \(z=x^2+y^2\):

For each $x, y, z$:

- Set $k$ equal to the variable and substitute it into the equation
- Sketch a two-dimensional graph with constant values of $k$ (e.g., $k=-2, -1, 0, 1, 2$) using the other two variables as axes

Combine the three **contour plots** in a three-dimensional plane to form the full sketch.

A hyperbola is formed when the difference between two points is constant. Where \(r\) is the x-intercept:

\[x^2-y^2=r^2\]

If \(r^2\) is negative, the hyperbola is is bounded by functions of \(x\), instead.

Limits of two-variable functions

A function is continuous at \((x, y)\) if and only if all possible lines through \((x, y)\) have the same limit. Or, where \(L\) is a constant:

\[\text{continuous}\iff \lim_{(x, y)\to(x_0, y_0)}f(x, y) = L\]

In practice, this means that if any two paths result in different limits, the limit is undefined. Substituting \(x|y=0\) or \(y=mx\) or \(x=my\) are common solutions.

!!! example For the function \(\lim_{(x, y)\to (0,0)}\frac{x^2}{x^2+y^2}\):

Along $y=0$:

$$\lim_{(x,0)\to(0, 0)} ... = 1$$

Along $x=0$:

$$\lim_{(0, y)\to(0, 0)} ... = 0$$

Therefore the limit does not exist.

Partial derivatives

Partial derivatives have multiple different symbols that all mean the same thing:

\[\frac{\partial f}{\partial x}=\partial_x f=f_x\]

For two-input-variable equations, setting one of the input variables to a constant will return the derivative of the slice at that constant.

By definition, the partial derivative of \(f\) with respect to \(x\) (in the x-direction) at point \((a, B)\) is:

\[\frac{\partial f}{\partial x}(a, B)=\lim_{h\to 0}\frac{f(a+h, B)-f(a, B)}{h}\]

Effectively:

  • if finding \(f_x\), \(y\) should be treated as constant.
  • if finding \(f_y\), \(x\) should be treated as constant.

!!! example With the function \(f(x,y)=x^2\sqrt{y}+\cos\pi y\):

\begin{align*}
f_x(1,1)&=\lim_{h\to 0}\frac{f(1+h,1)-f(1,1)} h \\
\tag*{$f(1,1)=1+\cos\pi=0$}&=\lim_{h\to 0}\frac{(1+h)^2-1} h \\
&=\lim_{h\to 0}\frac{h^2+2h} h \\
&= 2 \\
\end{align*}

Higher order derivatives

!!! definition - wrt. is short for “with respect to”.

\[\frac{\partial^2f}{\partial x^2}=\partial_{xx}f=f_{xx}\]

Derivatives of different variables can be combined:

\[f_{xy}=\frac{\partial}{\partial y}\frac{\partial f}{\partial x}=\frac{\partial^2 f}{\partial xy}\]

The order of the variables matter: \(f_{xy}\) is the derivative of f wrt. x and then wrt. y.

Clairauts theorem states that if \(f_x, f_y\), and \(f_{xy}\) all exist near \((a, b)\) and \(f_{yx}\) is continuous at \((a,b)\), \(f_{yx}(a,b)=f_{x,y}(a,b)\) and exists.

!!! warning In multivariable calculus, differentiability does not imply continuity.